Generalized Investigated in Topological Space

Abstract

: In this paper, some characterizations and proportion of notion a investigated. Throughout this paper (X, ?) and (Y, ? ) (simply, X and Y) represent topological spaces on which separation axioms are assumed unless otherwise mentioned. We introduce a new class of sets called regular generalized open sets which is properly placed in between the class of open sets and the class of - open sets. Throughout this paper (X, ?) represents a topological space on which no separation axiom is assumed unless otherwise mentioned. For a subset A of a topological space X, cl (A) and int (A) denote the closure of A and the interior of A respectively. X/A or Ac denotes the complement of A in X. introduced and investigated semi open sets, generalized closed sets, regular semi open sets, weakly closed sets, semi generalized closed sets , weakly generalized closed sets, strongly generalized closed sets, generalized pre - regular closed sets, regular generalized closed sets, and generalized ?-generalized closed sets respectively.

Introduction

Conclusion

A topological space (X, ?) is said to be ?*-connected if it is not the union of two nonempty disjoint ?*- open sets. If (X, ?) is a ?* - connected space and f:(X, ?) ? (Y, ?) has a (?*, ?)-graph and ?*- continuous function, the constant. Suppose that f is not constant. There exist disjoint points x, y ? X such that f(x) = f(y). Since (x, f(x)) G (f), then y ? f(x), hence by, there exist open sets U and V containing x and f(x) respectively such that f(U) ? V = ?. Since f is ? * - continuous, there exist a ? * - open sets G containing y such that f(G) ? V. Since U and V are disjoint ?*- open sets of (X, ?), it follows that (X, ?) is not ?*- connected Therefore, f is constant. Let (X1, ?1), (X2, ?2) and (X, ?) be topological spaces. Define a function f: (X, ?) ? (X1 × X2, ?1 × ?2) by f(x) = (f(x1), f(x2)). Then fi: X ? (Xi, ?i), where (i = 1, 2) is ?*-continuous if f is ?*- continuous.

References

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Copyright

Copyright © 2023 Kharatti Lal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.